Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces

This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

The strong nil-cleanness of semigroup rings

In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

On sturdy frame of abstract algebras

We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an application, we study the sturdy distributive lattice of lattice ordered groups.

KNITTED SEMILATTIC STRUCTURE OF REGULAR ${\cal H}^\dag$-CYBER GROUPS

We consider regular [Formula: see text]-cyber groups in the class of [Formula: see text]-abundant semigroups. By using knitted semilattice of semigroups, we give some structure theorems for regular [Formula: see text]-cyber groups, right quasi-normal [Formula: see text]-cyber groups and normal [Formula: see text]-cyber groups. Our main result generalizes a classical theorem of Petrich- Reilly on normal cryptic groups from the class of regular semigroups to the class of generalized abundant semigroups and also entriches a recent result of Guo-Shum on left cyber groups.

A Structure Theorem of Regular ${\cal H}^\sharp$-Cryptographs

A new set of generalized Green relations is given in studying the [Formula: see text]-abundant semigroups. By using the generalized strong semilattice of semigroups recently developed by the authors, we show that an [Formula: see text]-abundant semigroup is a regular [Formula: see text]-cryptograph if and only if it is an [Formula: see text]-strong semilattice of completely [Formula: see text]-simple semigroups. This result not only extends the well known result of Petrich and Reilly from the class of completely regular semigroups to the class of semiabundant semigroups, but also generalizes a well known result of Fountain on superabundant semigroups from the class of abundant semigroups to the class of semiabundant semigroups.

On Refined Semilattices of Semigroups

We give a short proof for the associativity of the binary operation defined on a refined system of semigroups indexed by a semilattice. The main result given by Zhang, Shum and Zhang in 2001 on the refined semilattice of semigroups is substantially strengthened.

Strong Semilattices of Implicative Semigroups

It is well known that a strong semilattice of semigroups is still a semigroup. However, it is not known whether the analogous result carries or not for a strong semilattice of implicative semigroups. The main difficulty is that the implicative operation ∗ is hard to define on the semilattice, especially we need the operation ∗ of the implicative semigroup to be compatible with the negatively ordered relation and the semigroup multiplication on the semilattice of implicative semigroups. In this paper, we provide a practical method of constructing such a strong semilattice of mplicative semigroups. Our method is particularly useful in constructing implicative semigroups of large size. A constructed example is given.

Completely Regular Semigroups with Generalized Strong Semilattice Decompositions

The concept of ρG-strong semilattice of semigroups is introduced. By using this concept, we study Green's relation ℋ on a completely regular semigroup S. Necessary and sufficient conditions for S/ℋ to be a regular band or a right quasi-normal band are obtained. Important results of Petrich and Reilly on regular cryptic semigroups are generalized and enriched. In particular, characterization theorems of regular cryptogroups and normal cryptogroups are obtained.